Solvent-averaged forces

The thennodynamic properties are calculated from the ion-ion pair correlation fimctions by generalizing the expressions derived earlier for one-component systems to multicomponent ionic mixtures. For ionic solutions it is also necessary to note that the interionic potentials are solvent averaged ionic potentials of average force ... 

The model is a McMillan-Mayer (MM)-level Hamiltonian model. Friedman characterizes models of this type as follows With MM-models it is interesting to see whether one can get a model that economically and elegantly agrees with all of the relevant experimental data for a given system success would mean that we can understand all of the observations in terms of solvent-averaged forces between the ions. However, it must be noted that there is no reason to expect the MM potential function to be nearly pairwise additive. There is an upper Imund on the ion concentration range within which it is sensible to compare the model with data for real systems if the pairwise addition approximation is made. ... 

In this section we consider two ions a and b, of arbitrary species but assumed for simplicity to be spherical, at an arbitrary but fixed center-to-center distance r in a large mass of solvent. The potential Uahir) of the force between the ions is called a solvent-averaged potential. Its relation to the intermolecular forces in a model at the BO level is given in Section 4 but it has some features that can be discussed here in a less formal way. 

It should be noticed that for the evaluation of dynamical properties, Mac-Millan Mayer hamiltonian is not sufficient, since it does not include the solute-solvent forces which are missing in a solvent averaged model. 

As in Eq (1), f(t) is a Gaussian random force, but now, the angular friction coefficient Cr is proportional to the solvent-averaged time correlation function of f(t) ... 

The theoretical approach based on the HNC integral equation is described in the context of ionic specificity. Two levels of description of the water medium are considered. Within the Primitive Model (continuous solvent), ionic specificity is introduced via effective, solvent-averaged, dispersion forces. The agreement with experimental data in bulk or at air-water interfaces is only partial and illustrates the limits of that approach. Within the Born-Oppenheimer model, the molecular HNC equation is solved with an explicit description of the solvent molecules (SPC water). Ionic and solvent profiles in bulk and at interfaces are enriched by short-range osdUated structures. The ionic polaris-ability is introduced via the self-consistent mean-field theory, the polarisable ions carrying an effective, fixed dipole moment. The study of the air-water interface reveals the limits of the conventional HNC approach and the needs for improved integral equations. 

At low solvent density, where isolated binary collisions prevail, the radial distribution fiinction g(r) is simply related to the pair potential u(r) via g ir) = exp[-n(r)//r7]. Correspondingly, at higher density one defines a fiinction w r) = -kT a[g r). It can be shown that the gradient of this fiinction is equivalent to the mean force between two particles obtamed by holding them at fixed distance r and averaging over the remaining N -2 particles of the system. Hence w r) is called the potential of mean force. Choosing the low-density system as a reference state one has the relation... 

Colloidal particles can be seen as large, model atoms . In what follows we assume that particles with a typical radius <3 = lOO nm are studied, about lO times as large as atoms. Usually, the solvent is considered to be a homogeneous medium, characterized by bulk properties such as the density p and dielectric constant t. A full statistical mechanical description of the system would involve all colloid and solvent degrees of freedom, which tend to be intractable. Instead, the potential of mean force, V, is used, in which the interactions between colloidal particles are averaged over... 

One approach to this problem is to use a potential of mean force (PMF), which describes he the free energy changes as a particular coordinate (such as the separation of two atoms or t torsion angle of a bond) is varied. The free energy change described by the potential of me force includes the averaged effects of the solvent. 

It is important to verify that the simulation describes the chemical system correctly. Any given property of the system should show a normal (Gaussian) distribution around the average value. If a normal distribution is not obtained, then a systematic error in the calculation is indicated. Comparing computed values to the experimental results will indicate the reasonableness of the force field, number of solvent molecules, and other aspects of the model system. 

A chemical will be a solvent for another material if the molecules of the two materials are compatible, i.e. they can co-exist on the molecular scale and there is no tendency to separate. This statement does not indicate the speed at which solution may take place since this will depend on additional considerations such as the molecular size of the potential solvent and the temperature. Molecules of two different species will be able to co-exist if the force of attraction between different molecules is not less than the forces of attraction between two like molecules of either species. If the average force of attraction between dissimilar molecules A and B is and that between similar molecules of type B Fbb and between similar molecules of type A F a then for compatibility Fab - bb and AB - P/KA- This is shown schematically in Figure 5.5 (a). 

Whereas the quasi-chemical theory has been eminently successful in describing the broad outlines, and even some of the details, of the order-disorder phenomenon in metallic solid solutions, several of its assumptions have been shown to be invalid. The manner of its failure, as well as the failure of the average-potential model to describe metallic solutions, indicates that metal atom interactions change radically in going from the pure state to the solution state. It is clear that little further progress may be expected in the formulation of statistical models for metallic solutions until the electronic interactions between solute and solvent species are better understood. In the area of solvent-solute interactions, the elastic model is unfruitful. Better understanding also is needed of the vibrational characteristics of metallic solutions, with respect to the changes in harmonic force constants and those in the anharmonicity of the vibrations. 

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